A double moving average control chart: Discussion

Abstract A double moving average (DMA) control chart has been proposed in the literature for monitoring the process mean. Several studies have also been done based on this scheme. Unfortunately, the variance of DMA statistic that has been used in these studies is not correct. In this article, we provide the correct variance of the DMA chart and through a simulation study, we evaluate its performance. It is shown that the DMA chart is more effective than the moving average (MA) chart in detecting small shifts in the process mean and vice versa for moderate to large shifts. Moreover, the DMA chart, even with a small value of span w, has similar, and in some cases better, detection ability than the EWMA and CUSUM charts, especially for moderate and large shifts.


Introduction
Control charts are the most important tool of Statistical Process Control (SPC) and are used to monitor the location or dispersion parameter. Shewhart (1926) first introduced the control chart technique and since then, control charts have been used in many manufacturing processes. Although the Shewhart control charts are very easy to use and also very effective in detecting large shifts, they are ineffective in detecting small to moderate shifts. For this purpose, Page (1954) developed the cumulative sum (CUSUM) chart and Roberts (1959Roberts ( , 1966 proposed the moving average (MA) and the exponentially weighted moving average (EWMA) control charts. These charts are memory-type as their charting statistics are based not only on the current information, but also on the past.
The MA control chart with span w is based on monitoring the average of w most current observations. It is less sensitive than the EWMA and CUSUM charts in detecting small to moderate shifts, to moderate shifts. Many authors, such as Areepong and Sukparungsee (2011), Areepong (2016), Phant, Sukparungsee, and Areepong (2016), Bualuang, Areepong, and Sukparungsee (2017), Aslam et al. (2017), Phantu, Sukparungsee, and Areepong (2018) and Adeoti, Akomolafe, and Adebola (2019), have studied the DMA control chart. Unfortunately, the variance of the DMA statistic, computed by Khoo and Wong (2008), was not correct. Thus, the results of previous studies are unreliable.
In the present article, we compute the correct variance of the DMA statistic and we also study the DMA chart de novo. The rest of this article is organized as follows. In Sec. 2, we give a brief description of the DMA control chart as proposed by Khoo and Wong (2008) while in Sec. 3, we compute the correct variance of the DMA statistic in order to develop the DMA chart with correct control limits. In Sec. 4, we evaluate via Monte Carlo simulations the performance of the DMA chart and in Sec. 5, we compare it with the MA, EWMA and CUSUM charts. An illustrative example with a real dataset is provided in Sec. 6. Finally, conclusions are summarized in Sec. 7.
2. The double moving average control chart by Khoo and Wong (2008) Suppose that X ij , with i ¼ 1, 2, :::m and j ¼ 1, 2, :::n is the jth observation in the ith sample or subgroup of size n ! 1: Moreover, it is assumed that the observations X ij are independent and identically distributed random variables and the underlying process is normally distributed, i.e., X ij $ iid Nðl 0 , r 2 Þ, where l 0 and r 2 are the in-control (IC) process mean and variance, respectively.
It is well-known that the subgroup averages X 1 , X 2 , ::: are computed by X i ¼ ðX i1 þ X i2 þ ::: þ X in Þ=n, i ¼ 1, 2, :::, m, and follow the normal distribution with mean and variance equal to l 0 and r 2 =n, respectively.
The moving average statistic MA i of span w at time i is defined as (Montgomery 2013) The mean or expected value of the statistic MA i is l 0 and its variance is given by Khoo and Wong (2008) proposed the double moving average (DMA) control chart which is based on computing the MA of subgroup averages twice. The calculation of the DMA i statistic of span w at time i is defined as They showed that the mean value of DMA i statistic for i ! w is equal to which is the same as for periods i < w. They also calculated the variance of DMA i statistic and they found that for w > 2 is given by , for i ! 2w À 1: For the case where w ¼ 2, they noted that the variance of DMA i statistic is obtained by using the first and third branches of Equation (5). As a result, they defined the control limits of the DMA chart for w > 2 as follows where L > 0 is the width of control limits. When w ¼ 2, the control limits of the DMA chart are given by the first and third branch of Equation (6). The centerline (CL) of the DMA chart is the IC value of process mean l 0 . The DMA chart is constructed by plotting the DMA i statistics versus the sample number or time i. If a charting statistic exceeds the control limits, then the process is considered to be out-of-control (OOC). Otherwise, the process is IC.

The corrected control limits of the DMA chart
Unfortunately, the mathematical expression of the variance of DMA i statistic reported in Equation (5) is not correct as the terms of covariance between MA i statistics have been ignored. For example, assume that w ¼ 4 and i ¼ 3. Then, as per definition Therefore, On the other hand, according to the formula given in the first branch of Equation (5), we have VarðDMA 3 Þ ¼ 11r 2 54n : As we will prove later, the added term of 14r 2 54n corresponds to the covariances between MA i statistics.
In the following lines, we compute the mean and variance of DMA i statistic.
For i < w, we have where a l ¼ P i j¼l 1=j: Therefore, for i < w, we have Now, for i < w, the mean value of DMA i statistic is Also, the variance of DMA i statistic is We can simplify P i l¼1 a 2 l as follows So, an alternative derivation of VarðDMA i Þ is given by But, for 1 j i, and, for 1 j 1 < j 2 i, Therefore, the variance of DMA i statistic is computed by Back to the previous example, as per our formula, we have a 1 ¼ 11=6, a 2 ¼ 5=6, a 3 ¼ 1=3 and as a result For w i < 2w À 1, the DMA i statistic is defined as Its mean value is, and its variance is computed by Now, Therefore, we have Thus, for w i < 2w À 1, the variance of DMA i statistic is given by For i ! 2w À 1, the DMA i statistic is defined as The mean value of DMA i statistic is and its variance is computed by and, for i À w þ 1 j 1 < j 2 i, Therefore, for i ! 2w À 1, the variance of DMA i statistic is given by Thus, we may conclude that, for any i, the mean value of DMA i statistic is equal to l 0 and about its variance, we have for i < w for w i < 2w À 1, and for i ! 2w À 1 In the "Online Supplement", we compute the variance of DMA i statistic for span w ¼ 5.
The corrected control limits of the DMA chart are given by where VarðDMA i Þ is given by Equation (7) if i < w, Equation (8) if w i < 2w À 1 and Equation (9) if i ! 2w À 1: When w ¼ 2, the control limits of the DMA chart are computed based on Equations (7) and (9).

Performance evaluation of DMA chart
The performance of a control chart is usually measured in terms of its run-length distribution. The run-length is the number of statistics that must be plotted in a control chart before initiating an OOC signal. The average run-length (ARL) is defined as the expected number of charting statistics that must be plotted before a statistic indicates an OOC signal (Montgomery 2013). When the process is IC, one would like to have a large value of ARL (ARL 0 ), so that the control chart signal a false alarm as slow as possible. On the other side, when the process is OOC, one would like to have a small value of ARL (ARL 1 ), so that the control chart detect the shift quickly. In this article, we use the ARL and the standard deviation of run-length (SDRL) performance measures. The run-length distribution of the DMA chart is calculated performing Monte Carlo simulations in R with 10,000 repetitions. Without loss of generality, we consider that the underlying process for the IC condition follows the N(0, 1) distribution. Table 1 presents the L values of DMA charts with w ¼ 2, 3, 4, 5, 8, 10, 12 and 15 for different ARL 0 values. It can be seen that as the value of w increases, the value of L decreases in order to obtain the desired ARL 0 value.
To study the performance of the DMA chart, we use some of L values shown in Table 1. We point out that the zero-state ARL performance is studied, as it is assumed that the shift occurs at the start. As the normal distribution is symmetric, we consider only positive shifts (in units of standard deviation) of the process mean; these are d ¼ 0:2, 0:4, 0:6, 0:8, 1:0, 1:25, 1:5, 2:0 and 3.0. The results are similar for negative shifts. Moreover, the performance is evaluated for the case of individual measurements (n ¼ 1) and subgrouped data of size n ¼ 5.  Tables 2 and 3 display the ARL and SDRL (in the parenthesis) values given an ARL 0 % 200 and 370, respectively. The results clearly show that the performance of the DMA chart is better as the value of w increases. It is also noted that the new results based on the correct control limits indicate that the DMA chart is less effective in detecting small shifts, but more effective in detecting moderate to large shifts than the initial study of DMA chart (Khoo and Wong 2008).

Comparison study
To compare the performance of control charts, it is recommended to have a similar desired value of ARL 0 . The chart with the smaller ARL 1 value in a specific shift can detect it more quickly than the other charts. In this section, we compare the performance of the DMA chart with the MA, EWMA and CUSUM charts using the zero-state ARL measure. A brief description of these charts is given in the following lines.
The MA chart is based on plotting the MA statistic given by Equation (1) and its control limits are given by The process is considered to be OOC if any plotted point MA i exceeds the control limits. The EWMA chart in based on plotting the statistic where 0 < k 1 is the smoothing constant and Z 0 ¼ l 0 : The control limits of the EWMA chart are given by For large values of i, the control limits become  The classical Shewhart chart is a special case of the EWMA chart for k ¼ 1. Small values of k are recommended for detecting small shifts while larger values are more appropriate for detecting larger shifts (Montgomery 2013). A process is considered to be OOC if a plotted point Z i lies outside the control limits.
The charting statistics of a CUSUM chart are computed by where C þ 0 ¼ C À 0 ¼ 0 and K is the reference value, usually computed by  The CUSUM chart that has been designed to detect quickly a shift d is very effective for the specific shift. A process is considered to be OOC if a charting statistic exceeds the decision interval , where h > 0. Otherwise, the process is considered to be IC. To make valid conclusions, the ARL 0 value of the competing control charts is set equal to 370. For the MA chart, we use the same values of w with those of DMA chart while for the EWMA chart, we use the asymptotic control limits and values of k ¼ 0:05, 0:10, 0:25, 0:50, 0:75 and 1.00, where the latter value corresponds to the Shewhart chart. Finally, the CUSUM chart is designed to detect quickly shifts of d ¼ 0:2, 0:8 and 1.5. The OOC performance of MA, EWMA and CUSUM charts is presented in Tables 4-6, respectively.
Comparing the DMA and MA charts for individual measurements, we conclude that the range of shifts where the DMA chart outperforms the MA chart decreases as the value of w increases. For example, the DMA chart is more effective than the MA chart in detecting shifts of d 2:0 when w ¼ 2, d 1:0 when w ¼ 5 and d 0:6 when w ¼ 10, 12 or 15. For the rest range of shifts, i.e., for larger shifts, the MA chart is more effective than the DMA chart. On the other hand, when n ¼ 5, the DMA chart is still more sensitive in detecting small shifts, but its superiority  Tables 3 and 5 indicate that for the case of individual measurements, the EWMA chart with k ¼ 0:05 is more sensitive in detecting small shifts (d 0:6) while the DMA chart has comparable performance with the best-performing EWMA chart for shifts of 0:6 d 1:5: Additionally, as the size of shift in the specific range increases, a DMA chart with a small value of w performs similarly with the best-performing EWMA chart. For example, a DMA chart with w ! 10 has similar performance with the EWMA chart with k ¼ 0:10 for d ¼ 0:8 while a DMA chart with w ! 3 has similar detection ability with the EWMA chart with k ¼ 0:25 for d ¼ 1:5: Finally, the DMA chart is more sensitive than the EWMA chart for large shifts (d ! 2:0). For the case of subgrouped data, the differences between the ARL 1 values of the DMA chart with w ¼ 15 and the best-performing EWMA chart are negligible for small shifts (d 0:4) while a DMA chart with w ! 10 performs a slightly better than the best-performing EWMA chart for moderate shifts (0:8 d 1:5). For large shifts (d ! 2:0), the DMA chart performs similarly with EWMA charts with a large value of k. We point out that the DMA chart outperforms the Shewhart chart over the entire range of shifts.
A performance comparison between DMA and CUSUM charts when n ¼ 1 indicates that the CUSUM chart optimal designed to detect quickly a small shift is more effective than the DMA chart only for a small range of shifts around it while for moderate to large shifts, the DMA chart is more effective. A DMA chart with a large value of w is more sensitive than a CUSUM chart optimal designed to detect quickly a moderate or large shift for the entire range of shifts. For example, the CUSUM chart optimal designed to detect a shift of d ¼ 0:2 outperforms the DMA chart at this shift. For shifts 0:4 d 1:0, a DMA chart even with a small value of w is more effective than the specific CUSUM chart while for shifts d > 1:0, all DMA charts outperform the CUSUM chart. On the other side, a CUSUM chart optimal designed to detect quickly a shift of d ¼ 0:8 or 1.5 has the same or worse detection ability than a DMA chart with a large value of w at the specific shifts. For the case of subgrouped data, the DMA chart performs better even for shifts where the CUSUM chart is optimal designed. For example, a DMA chart with w ! 12 outperforms the CUSUM chart optimal designed for a shift of d ¼ 0:2 at the specific shift while all DMA charts are more sensitive for the rest range of shifts.

Illustrative examples
In order to demonstrate the practical application of the DMA chart, we use the inside diameter measurements (in mm) on automobile engine piston rings, provided Montgomery (2013). The dataset consists of 40 samples, each of size n ¼ 5 and is presented in Table 7. The first 25 samples represent the phase I observations. The IC values of process mean and standard deviation are l 0 ¼ 74:0012 mm and r ¼ 0:01 mm, respectively. The last 15 samples represent the OOC observations. Setting an ARL 0 ¼370, we construct a DMA chart with w ¼ 4 and L ¼ 2.780 (see Table 1). The charting statistics and the control limits are also presented in Table 7. The control chart is displayed in Figure 1. Moreover, we construct the MA chart with w ¼ 4, shown in Figure 2. From these Figures, we conclude that both charts detect a first OOC signal at the 27th sample.

Conclusion
The DMA control chart was proposed by Khoo and Wong (2008) in order to improve the performance of the MA control chart for the detection of small to moderate shifts. Unfortunately, the computed variance of the DMA statistic was not correct as the covariance between MA statistics was ignored. In this article, we calculate the correct variance of the DMA statistic and we study the performance of DMA chart performing numerical simulations. It is shown that the detection ability of DMA chart improves as the value of the span w increases. A comparison study with the MA chart demonstrates that the DMA chart is more effective for small to moderate shifts while this range of shifts becomes narrower as the value of w increases. Furthermore, for individual measurements, the DMA chart has similar performance with the EWMA and CUSUM charts for moderate shifts, but it is more (less) effective in detecting large (small) shifts. The superiority of the DMA chart versus the EWMA and CUSUM charts enlarges for subgrouped data.
In terms of future work, the researchers who have studied the DMA scheme can update their results using the correct control limits.